[Analysis/Topology] Issue with a proof

I have an exam tomorrow, and I'm going through all the theorems/proofs from my notes and I have this written:

From the proof of the "<=" direction of this theorem, f: S -> T is cont. on S iff for every open set Y ⊂ T, we have f^-1(Y) is open.

This is what I have written: Given f^-1 (Y) is open for every open Y. Prove f is cont.

Let p ∈ S, Ɐε > 0, B(f(p);ε) is open. From the given, f^-1 (B(f(p);ε)) is open. In particular, note that p∈ f^-1 (B(f(p);ε)) =>∃δ > 0 s.t. B(p;δ) ⊂ f^-1 (B(f(p);ε)) => f(B(f(p);ε)) ⊂ f(f^-1 (B(f(p);ε))) ⊂ B(f(p);ε) => f is cont. at p Ɐp ∈ S.

I'm confused about this part: f(B(f(p);ε)) ⊂ f(f^-1 (B(f(p);ε))) ⊂ B(f(p);ε). I'm hoping I just copied it wrong and it should just be f(B(p;δ)) ⊂ f(f^-1 (B(f(p);ε))) ⊂ B(f(p);ε) and I'm not missing something.

Is that what's wrong?